cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A322245 a(n) = A322244(n)^2, the square of the central coefficient in (1 + 3*x + 16x^2)^n.

Original entry on oeis.org

1, 9, 1681, 99225, 11189025, 1019077929, 108167974321, 11137724806329, 1197175570050625, 129043286745662025, 14139849178569444561, 1560056939689534791129, 173632800433761049827681, 19441545948416137605093225, 2189471746316530595782640625, 247739375533243457859858956025, 28151322919072607132343448452225, 3210745922813847247161044673062025
Offset: 0

Views

Author

Paul D. Hanna, Dec 09 2018

Keywords

Comments

The g.f. of A322244 is 1/sqrt(1 - 6*x - 55*x^2).

Examples

			G.f.: A(x) = 1 + 9*x + 1681*x^2 + 99225*x^3 + 11189025*x^4 + 1019077929*x^5 + 108167974321*x^6 + 11137724806329*x^7 + 1197175570050625*x^8 + ...
that is,
A(x) = 1 + 3^2*x + 41^2*x^2 + 315^2*x^3 + 3345^2*x^4 + 31923^2*x^5 + 328889^2*x^6 + 3337323^2*x^7 + 34600225^2*x^8 + ... + A322244(n)^2*x^n + ...

Links

Crossrefs

Cf. A322244.

Programs

  • Mathematica
    a[n_] := Sum[11^(n-k) * (-4)^k * Binomial[n,k] * Binomial[2k,k], {k, 0, n}]^2; Array[a, 20, 0] (* Amiram Eldar, Dec 13 2018 *)
  • PARI
    /* a(n) = A322244(n)^2 - g.f. */
{a(n)=polcoeff(1/sqrt(1 - 6*x - 55*x^2 +x*O(x^n)), n)^2}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* a(n) = A322244(n)^2 - binomial sum */
{a(n) = sum(k=0,n, 11^(n-k)*(-4)^k*binomial(n,k)*binomial(2*k,k))^2}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* Using AGM: */
{a(n)=polcoeff( 1 / agm(1 + 5*11*x, sqrt((1 - 5^2*x)*(1 - 11^2*x) +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f.: 1 / AGM(1 + 5*11*x, sqrt((1 - 5^2*x)*(1 - 11^2*x)) ), where AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) is the arithmetic-geometric mean.
G.f.: 1 / AGM((1-5*x)*(1-11*x), (1+5*x)*(1+11*x)) = Sum_{n>=0} a(n)*x^(2*n).
a(n) = A322244(n)^2, where A322244(n) = a(n) = Sum_{k=0..n} 11^(n-k) * (-4)^k * binomial(n,k)*binomial(2*k,k).
a(n) ~ 11^(2*n + 1) / (16*Pi*n). - Vaclav Kotesovec, Dec 13 2018

A307695 Expansion of 1/(sqrt(1-4*x)*sqrt(1-16*x)).

Original entry on oeis.org

1, 10, 118, 1540, 21286, 304300, 4443580, 65830600, 985483270, 14869654300, 225759595348, 3444812388280, 52781007848284, 811510465220920, 12513859077134008, 193460383702061200, 2997463389599395270, 46532910920993515900, 723626591914643806180, 11270311875128088314200
Offset: 0

Views

Author

Seiichi Manyama, Apr 22 2019

Keywords

Comments

Let 1/(sqrt(1-c*x)*sqrt(1-d*x)) = Sum_{k>=0} b(k)*x^k.
b(n) = Sum_{k=0..n} c^(n-k) * e^k * binomial(n,k) * binomial(2*k,k) = Sum_{k=0..n} d^(n-k) * (-e)^k * binomial(n,k) * binomial(2*k,k), where e = (d-c)/4.
n*b(n) = (c+d)/2 * (2*n-1) * b(n-1) - c * d * (n-1) * b(n-2) for n > 1.

Links

Crossrefs

Cf. A000984 (c=0,d=4,e=1), A026375 (c=1,d=5,e=1), A081671 (c=2,d=6,e=1), A098409 (c=3,d=7,e=1), A098410 (c=4,d=8,e=1), A104454 (c=5,d=9,e=1).
Cf. A084605 (c=-3,d=5,e=2), A098453 (c=-2,d=6,e=2), A322242 (c=-1,d=7,e=2), A084771 (c=1,d=9,e=2), A248168 (c=3,d=11,e=2).
Cf. A322246 (c=-1,d=11,e=3), this sequence (c=4,d=16,e=3).
Cf. A322244 (c=-5,d=11,e=4), A322248 (c=-3,d=13,e=4).

Programs

  • Mathematica
    a[n_] := Sum[4^(n-k) * 3^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]; Array[a, 20, 0] // Flatten (* Amiram Eldar, May 13 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(1/sqrt(1-20*x+64*x^2))
  • PARI
    {a(n) = sum(k=0, n, 4^(n-k)*3^k*binomial(n, k)*binomial(2*k, k))}
  • PARI
    {a(n) = sum(k=0, n, 16^(n-k)*(-3)^k*binomial(n, k)*binomial(2*k, k))}

Formula

a(n) = Sum_{k=0..n} 4^(n-k)*3^k*binomial(n,k)*binomial(2k,k).
a(n) = Sum_{k=0..n} 16^(n-k)*(-3)^k*binomial(n,k)*binomial(2k,k).
D-finite with recurrence: n*a(n) = 10*(2*n-1)*a(n-1) - 64*(n-1)*a(n-2) for n > 1.
a(n) ~ 2^(4*n+1) / sqrt(3*Pi*n). - Vaclav Kotesovec, Apr 30 2019
Showing 1-2 of 2 results.

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